Problem 64 - Algebra A Combined Function

Question

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. In \(2006,\) the U.S. population per square mile of land area was approximately \(83.6 .\) In 2000 , the population per square mile was 79.6 a. Assume the relationship between years past 2000 and population per square mile is linear over this period. Write an equation describing the relationship between year and population per square mile. Use ordered pairs of the form (years past 2000 , population per square mile). b. Use this equation to predict the population per square mile in 2010 .

Step-by-Step Solution

Answer

Equation: \(y = \frac{2}{3}x + 79.6\). Prediction for 2010: 86.27 people/sq. mile.

Step 1: Identify Ordered Pairs

We first need to establish our ordered pairs, where each pair is formed as (years past 2000, population per square mile). For the year 2000, the ordered pair is (0, 79.6). For 2006, it is (6, 83.6), as 2006 is 6 years past 2000.

Step 2: Calculate the Slope

The slope (m) of a line is calculated by \[m = \frac{y_2 - y_1}{x_2 - x_1}\\]Using our pairs (0, 79.6) and (6, 83.6), this becomes:\[m = \frac{83.6 - 79.6}{6 - 0} = \frac{4}{6} = \frac{2}{3}\\]

Step 3: Write the Equation in Slope-Intercept Form

The general form of a linear equation is given by \[y = mx + b\\]where \(m\) is the slope and \(b\) is the y-intercept. We can use the point (0, 79.6) to determine \(b\):\[79.6 = \left(\frac{2}{3}\right) \cdot 0 + b\\]Thus, \(b = 79.6\) and the equation is \[y = \frac{2}{3}x + 79.6\\]

Step 4: Predict Population per Square Mile for 2010

To predict the population per square mile in 2010, substitute \(x = 10\) into the equation \(y = \frac{2}{3}x + 79.6\):\[y = \frac{2}{3} \times 10 + 79.6 = \frac{20}{3} + 79.6 = 6.67 + 79.6 = 86.27\\]Thus, the predicted population per square mile in 2010 is 86.27.

Key Concepts

Linear Equation

A linear equation is a mathematical expression that describes a straight line when plotted on a graph. In its simplest form, this type of equation is expressed as \( y = mx + b \), where:

• \( y \) represents the dependent variable (often what we're solving for, like population density)
• \( m \) is the slope of the line, indicating the rate of change
• \( x \) is the independent variable, which usually represents time or another changing factor
• \( b \) is the y-intercept, the value of \( y \) when \( x \) equals zero

Linear equations are fundamental in understanding relationships in various fields such as economics, biology, and physics, since they model how one quantity changes in response to another. The simplicity of these equations makes them a powerful tool in both theoretical and real-world applications.

Ordered Pairs

Ordered pairs are a pair of numbers used to locate a point on a plane, typically represented as \((x, y)\). These pairs are crucial for plotting points on a graph and exploring the relationship between two numerical values.The first number in the pair, \(x\), is called the "abscissa," and it typically represents a horizontal position; the second number, \(y\), is called the "ordinate," representing the vertical position. Ordered pairs offer:

• A way to quantify changes over time
• An understanding of relationships between two numerical factors

In the context of the given problem, ordered pairs like \((0, 79.6)\) and \((6, 83.6)\) help establish the relationship between years (years since 2000) and population density values. They are the foundation for calculating slope and writing linear equations.

Population Density

Population density measures how many people live within a given land area, usually expressed as the number of people per square mile or square kilometer. This metric provides insight into how crowded or spacious an area is. In practical terms:

• High population density can indicate urban areas with bustling activities and limited space.
• Low density may suggest rural areas with more open space and luxuries like larger properties.

Understanding population density is crucial for infrastructure planning, resource distribution, and environmental impact analysis. In the exercise, population density increases incrementally as calculated, reflecting changes over time—an essential factor for urban planning and studying the demographic shifts.

Slope Calculation

The slope of a line on a graph indicates the steepness and direction of the line, and it is a measure of how much "rise" (change in \(y\)) occurs per unit of "run" (change in \(x\)). In mathematical terms, the slope \(m\) can be calculated using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here:

• \(x_1, y_1\) and \(x_2, y_2\) are coordinates of two distinct points on the line

For example, using the ordered pairs from the problem \((0, 79.6)\) and \((6, 83.6)\), the slope is calculated as \( \frac{4}{6} = \frac{2}{3} \). This indicates that for every year past 2000, the population density increases by \(\frac{2}{3}\) persons per square mile, indicating a gradual increase in population density. Understanding how to compute the slope allows us to predict future values and explore the trend's implications further.